The thermodynamic analysis presented here is an outline of the air-standard Brayton cycle and its various modifications. These modifications are evaluate to examine the effects they have on the basic cycle. One of the most important is the augmentation of power in a gas turbine. This is treated in a special section in this chapter. The Brayton Cycle
The Brayton cycle in its ideal form consists of two isobaric processes and two isentropic processes. The two isobaric processes consist of the combustor system of the gas turbine and the gas side of the HRSG. The two isentropic processes represent the compression (Compressor) and the expansion (Turbine Expander) processes in the gas turbine. Figure 2-1 shows the Ideal Brayton Cycle.
A simplified application of the first law of thermodynamics to the air-standar Brayton cycle in Figure 2-1 (assuming no changes in kinetic and potential energy has the following relationships: Work of compressor
Wc = ˙ma(h2 − h1) (2-1)
Work of turbine
Wt = ( ˙ma + ˙ mf )(h3 − h4) (2-2)
Total output work
Wcyc = Wt − Wc (2-3)
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Heat added to system
Q2,3 = ˙mf xLHVfuel = ( ˙ma + ˙ mf )(h3)− ˙ mah2 (2-4)
Thus, the overall cycle efficiency is
ηcyc = Wcyc/Q2,3 (2-5)
Increasing the pressure ratio and the turbine firing temperature increases the
Brayton cycle efficiency. This relationship of overall cycle efficiency is based on
certain simplification assumptions such as: (1) ˙ma ˙mf , (2) the gas is calorically
and thermally perfect, which means that the specific heat at constant pressure (cp)
and the specific heat at constant volume (cv) are constant thus the specific heat
ratio γ remains constant throughout the cycle, (3) the pressure ratio (rp) in both
the compressor and the turbine are the same, and (4) all components operate at
100% efficiency. With these assumptions the effect on the ideal cycle efficiency
as a function of pressure ratio for the ideal Brayton cycle operating between
the ambient temperature and the firing temperature is given by the following
relationship:
ηideal =
1 − 1
r
γ−1
γ
p
(2-6)
where Pr = Pressure Ratio; and γ is the ratio of the specific heats. The above
equation tends to go to very high numbers as the pressure ratio is increased.
Assuming that the pressure ratio is the same in both the compressor and the
turbine the following relationships hold using the pressure ratio in the compressor:
ηideal = 1 − T1
T2
(2-7)
and using the pressure ratio in the turbine
ηideal = 1 − T4
T3
(2-8)
In the case of the actual cycle the effect of the turbine compressor (ηc) and
expander (ηt ) efficiencies must also be taken into account to obtain the overall
cycle efficiency between the firing temperature Tf and the ambient temperature
Tamb of the turbine. This relationship is given in the following equation:
ηcycle =
ηtTf − Tambr
γ−1
γ
p
ηc
Tf − Tamb − Tamb
rp
γ−1
γ
− 1
ηc
1 − 1
r
γ−1
γ
p
(2-9)
Figure 2-2 shows the effect on the overall cycle efficiency of the increasing
pressure ratio and the firing temperature. The increase in the pressure ratio
increases the overall efficiency at a given firing temperature; however, increasing
the pressure ratio beyond a certain value at any given firing temperature can
actually result in lowering the overall cycle efficiency. It should also be noted
that the very high-pressure ratios tend to reduce the operating range of the turbine
compressor. This causes the turbine compressor to be much more intolerant
to dirt build-up in the inlet air filter and on the compressor blades and creates
large drops in cycle efficiency and performance. In some cases, it can lead to
Figure 2-2. Overall cycle efficiency as a function of the firing temperature and
pressure ratio. Based on a compressor efficiency of 87% and a turbine efficiency
of 92%.
compressor surge, which in turn can lead to a flameout, or even serious damage
and failure of the compressor blades and the radial and thrust bearings of the gas
turbine.
To obtain a more accurate relationship between the overall thermal efficiency
and the inlet turbine temperatures, overall pressure ratios, and output work,
consider the following relationships. For maximum overall thermal cycle efficiency,
the following equation gives the optimum pressure ratio for fixed inlet
temperatures and efficiencies to the compressor and turbine:
(rp)copt =
1
T1T3η1 − T1T3 + T 2
1
[T1T3ηt
−
(T1T3ηt )2 − (T1T3ηt − T1T3 + T 2
1 )(T 2
3 ηcηt − T1T3ηcηt + T1T3ηt )]
γ
γ−1
(2-10)
The above equation for no losses in the compressor and turbine (ηc = ηt = 1)
reduces to:
(rp)copt =
T1T3
T 2
1
γ
γ−1
(2-11)
The optimum pressure ratio for maximum output work for a turbine taking
into account the efficiencies of the compressor and the turbine expander section